polytonal jazz & dave brubeck

Jazz PerspectivesVol. 3, No. 2, August 2009, pp. 153–176

ISSN 1749–4060 print/1749–4079 online © 2009 Taylor & FrancisDOI: 10.1080/17494060903152396

Dave Brubeck and Polytonal JazzMark McFarland

Taylor and FrancisRJAZ_A_415412.sgm10.1080/17494060903152396Jazz Perspectives1749-4060 (print)/1749-4079 (online)Original Article2009Taylor & [email protected] Dave Brubeck is universally known as a jazz pianist, he describes himself as“a composer who plays the piano.”1 This distinction highlights the lesser-knownaspects of Brubeck’s career, including his formal training at both the University of thePacific and Mills College, as well as the numerous “serious” works he has composed.Because of Brubeck’s thorough training in both classical and jazz idioms, he has oftenbeen described as the first musician to fuse jazz and classical music successfully.2

The meeting of classical and jazz idioms is pervasive in Brubeck’s output. Forinstance, he famously introduced asymmetric meter into jazz with his 1959 albumTime Out, which includes the highly popular tunes “Take Five” (in quintuple meter)and “Blue Rondo à la Turk” (written in 9/8, but with an irregular grouping of beatdivisions: 2 + 2 + 2 + 3). Brubeck incorporated an even more radical aspect of twenti-eth-century music into jazz through his use of polytonality.3

Polytonality is a term that has yet to be defined to the satisfaction of all, and suchagreement will likely never happen.4 This impasse is due to the inherent contradictionof the term itself and the ambiguity of the musical technique. In its most literal inter-pretation, polytonality implies the simultaneous unfolding of multiple tonalities.5 Thisis, strictly speaking, an impossibility, which explains why Benjamin Boretz commentsthat polytonality embodies an indeterminate reference and Pieter van den Toorn morecolorfully describes it as “a real horror of the musical imagination.”6 Composers whowrote in this style and published articles on this topic in the early part of the twentiethcentury—most notably (although not exclusively) the French composer Darius

1 Richard Wang, “Dave Brubeck,” vol. 4, The New Grove Dictionary of Music and Musicians, 2nd ed., ed. StanleySadie (London: Macmillan, 2001), 452.2 George T. Simon, liner notes to Dave Brubeck: Greatest Hits, Columbia 32046, 1967, LP. Brubeck discussed thissubject on the radio show Piano Jazz. In this broadcast, the show’s host, Marian McPartland, mentioned that she“was interested to hear you [Brubeck] say that you approach jazz from the classics. You know a lot of people whodon’t believe that, do they? They think, ‘jazz and classics: never the twain shall meet.’” Brubeck responded “nevershall they part,” to which McPartland agreed. Marian McPartland’s Piano Jazz with Guest Dave Brubeck, JazzAlliance 12001, 1993, compact disc.3 This is a topic that has received little attention despite the fact that Brubeck has mentioned his use of polytonalityrepeatedly in interviews, including those for television. See, for example, Brubeck’s October 17, 1961, appearanceand interview on Ralph Gleason’s Jazz Casual, Wea Corporation DVD B00006RJCR, 2003.4 François de Médicis points out that questions over the definition of this term date back to the origins of this tech-nique. See François de Médicis, “Darius Milhaud and the Debate on Polytonality in the French Press of the 1920s,”Music & Letters 86 (2005): 576.5 See Allen Forte, Contemporary Tone Structures (New York: Teachers College, Columbia University, 1955), 137.6 See Benjamin Boretz, Meta-Variations: Studies in the Foundations of Musical Thought (New York: Open Space,1995), 243, and Pieter van den Toorn, The Music of Igor Stravinsky (New Haven: Yale University Press, 1983), 63.

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154 Dave Brubeck and Polytonal Jazz

Milhaud—did not define polytonality in the same literal way.7 Instead, to them, poly-tonality was what today would be referred to as a polychord: a verticality made up ofdistinct chords and partitioned to project this construction. There are problems,however, with even this loose definition. Can any chords heard together create a poly-chord? And does the perception of a polychord matter in its analysis? For example, theinsistence that both chords belong to separate diatonic collections would disallow theopening polychord from Aaron Copland’s Appalachian Spring (C# – E – A – B – E –G#). Yet if both chords can belong to the same diatonic collection, then any seventhchord could be defined as a polychord—for instance, G – B – D and B – D – F.“Polychords” of this level of simplicity are clearly heard as single chords (i.e., theyproject a single tonality or root), as are many polychords of much greater complexity.It is no wonder then that a consensus cannot be reached on the definition of polytonal-ity when each musician, based on their own perception and experience, draws a linesomewhere in the grey area of polychords, to separate exotic diatonic harmony frompolytonality.8

The major writings on polytonality from the 1920s mentioned above often implicitlyexclude mere seventh chords, while they frequently include polychords made of triadsfrom the same diatonic collection. The equation of polytonality with polychordsallowed these authors to trace a rich pre-history of polytonality, one that went back forMilhaud to J. S. Bach9 and for the French composer and writer, Charles Koechlin, backto the sixteenth century.10 It is obvious that the early examples of polychords fromthese two composers did not evoke multiple tonal centers. The literal definition of theterms polytonalité or polytonie11 does not therefore seem to have bothered any of thecomposer/theorists of this era.12 This paper will use the term polytonality in this sameway because this is the specific definition of the term that was used by Milhaud and

7 Stravinsky spoke of the second act of Petrushka as being written “in two keys.” See Igor Stravinsky and RobertCraft, Expositions and Developments (New York: Doubleday, 1962), 162. Ravel provided an analysis of a passagefrom his Valses nobles et sentimentales, demonstrating that polytonality can be formed by unresolved appoggiaturas.See René Lenormand, A Study of Twentieth-Century Harmony (London: Joseph Williams, 1915), 62–63. See alsoDarius Milhaud, “Polytonalité et atonalité,” La Revue Musicale 4 (1923): 29–44; Alfredo Casella, “Tone-Problemsof To-day,” Musical Quarterly 10 (1924): 159–71; and Charles Koechlin, “Évolution de l’harmonie: Périodecontemporaine depuis Bizet et César Franck jusqu’à nos jours,” vol. 2, Encyclopédie de la Musique et Dictionnairedu Conservatoire, edited by Lavignac and La Laurencie (Paris: Delagrave, 1925), 591–760, and Traité de l’harmonie(Paris: Eschig, 1927–30).8 All writers on polytonality make a distinction between harmonic and melodic polytonality. The former is themore common version involving polychords, while the latter is composed of a polyphonic texture, with each linewritten in a distinct tonality.9 Milhaud, “Polytonalité et atonalité,” 30–31.10 Koechlin, Traité de l’harmonie, 252.11 For a discussion of how these two terms were used synonymously in the 1920s, see de Médicis, “DariusMilhaud,” 574.12 For other recent attempts to define polytonality, see Daniel Harrison, “Bitonality, Pentatonicism, and Diatoni-cism in a Work by Milhaud,” in Music Theory in Concept and Practice, eds. James M. Baker, David W. Beach, andJonathan W. Bernard (Rochester, NY: University of Rochester Press, 1997), 393–408; Deborah Mawer, “In Pursuitof an Analytical Approach,” in Darius Milhaud: Modality & Structure in Music of the 1920s (Aldershot, UK:Ashgate, 1997), 18–56; Peter Kaminsky, “Ravel’s Late Music and the Problem of ‘Polytonality’,” Music TheorySpectrum 26 (2004): 237–264; and Barbara Kelly, “Polytonality, Counterpoint and Instrumentation,” in Traditionand Style in the Works of Darius Milhaud, 1912–1939 (Aldershot, UK: Ashgate, 2003), 142–168.


Jazz Perspectives 155

passed on to Brubeck. In this paper’s analyses, however, I have nevertheless remainedmindful of the distinction between simple polychords and the true projection ofmultiple tonalities, since Brubeck masterfully exploits the grey area between these tworealms.

This study will further explore the use of polytonality across Brubeck’s long career.As such, this sort of survey necessitates a discussion of the changing nature of hispolytonal writing. The main goal of this broad examination is to go beyond the mereidentification of the various keys that are superimposed—a simplistic analytic goalrightly derided by Daniel Harrison13—and explore both the various means used toachieve polytonality and the methods by which polytonality is incorporated intoBrubeck’s works. The latter objective is an especially important point since the majorityof the works analyzed below fuse polytonality with traditional jazz harmony.

The origins of Brubeck’s interest in polytonality date back to his introduction toMilhaud, with whom he studied at Mills College. Although it seems inevitable in hind-sight that these two men would work together, it was a happy coincidence that they metat all: while Brubeck was raised near the San Francisco Bay Area, Milhaud ended up inOakland by chance, only learning of his Mills College teaching appointment in thecourse of his Atlantic crossing.14 Milhaud was open to jazz; indeed, many of his earlycompositions incorporated jazz elements, and most famously in his ballet score Lacréation du monde (1923). In his Mills College years, Milhaud even went so far as toinvite his students to write their homework assignments as jazz compositions, andfrom this sort of instruction, the Dave Brubeck Octet and many of the compositions ofthis vibrant young musical circle were born.15

It was typical of Milhaud’s non-dogmatic approach to teaching that he incorporatedjazz into his courses in the 1940s, well after he admitted (in 1926) that he had lost inter-est in this music.16 He notably included polytonality in his Mills College courses; infact, Milhaud’s graduate composition students began their studies by being given acopy of Milhaud’s article on this subject.17 Milhaud’s students would no doubt havealready been aware of his compositions, however. One of Milhaud’s most popularworks, Scaramouche, from 1937, was likely such a piece. In fact, this is one of Brubeck’sfavorite compositions by the French composer for reasons that become quickly appar-ent after examination (see Example 1). Milhaud establishes C major in both pianos inmm. 1–3, although this key is used by only one of the pianos (see the second piano inm. 4, and then the first piano in m. 5) in the following two measures. The oppositepiano (the first piano in m. 4 and the second piano in m. 5) presents a chromatically-related series of triads that do not belong to any single key, but which consistently clash

13 Harrison, “Bitonality, Pentatonicism, and Diatonicism,” 394.14 Darius Milhaud, My Happy Life: An Autobiography (London: Marion Boyars Publishers, 1995), 201.15 Brubeck’s octet included several other performers and composers who would later establish their own signifi-cant reputations, including Paul Desmond, Cal Tjader, David van Kriedt (tenor saxophonist and composer of“Fugue on Bop Themes”), and Bill Smith (clarinetist and member of Brubeck’s current quartet).16 Milhaud, My Happy Life, 146.17 This was related to the author in a March 2003 conversation with Dr. Katherine Warne, a former student ofMilhaud (she is also currently president of the Milhaud Society). Milhaud, “Polytonalité et atonalité,” cited above.



with the diatonic triad with which they are paired. The partitioning of polytonal keysby register is a technique that Brubeck picked up in works like Scaramouche and contin-ues to use to this day, with one (or more) keys presented in both his left and righthands.

Brubeck did not need to be given a copy of “Polytonalité et atonalité.” As Brubeckhad met Milhaud while still an undergraduate at the University of the Pacific, he soughtout this article years before he began his graduate studies with the French composer.18

The impact of Milhaud’s article was immediately apparent in Brubeck’s playing. PaulDesmond remembered that Brubeck had confounded him on their first meeting byasking to perform the blues in G and then playing with his left hand in G and his righthand in Bb.19 Brubeck later demonstrated this effect during his appearance on theradio program, Piano Jazz, with Marian McPartland (see Example 2).20

18 This meeting was undoubtedly arranged by Brubeck’s older brother, Howard, who was one of Milhaud’s firstteaching assistants. Howard eventually substituted for Milhaud at Mills College during the latter’s semiannualleaves to teach at the Paris Conservatoire.19 Even more tellingly, Desmond recalls that by 1949 he sometimes had to ask Brubeck to simplify his playing sinceBrubeck would often accompany Desmond’s solos in three keys at once. See Marian McPartland, “Perils of Paul,”in All in Good Time (New York: Oxford University Press, 1987), 59.20 Marian McPartland’s Piano Jazz with Guest Dave Brubeck. Transcription of the example by the present author.

Example 1 Darius Milhaud, Scaramouche (1937), mm. 1–5. Copyright © Francis SalabertEditions.



The polytonality of passages in Brubeck like that shown in Example 2—where hesuperimposes minor-third related keys (a combination that he admits has become asnatural to him as playing with both hands in the same key)—present a perfect illustra-tion of how the notation of polytonality can be quite different from its perception.21 Inthis example, if each of the hands is played separately, the left-hand clearly establishesG as its tonic, while the right hand’s melody is obviously in Bb. Yet when the two handsare combined, it is difficult to hear both tonal centers clearly projected; instead, due toregistral prominence, the key of Bb is heard to be the tonic, while the foreign notesfrom G major are heard as a set of characteristic dissonances.22 Such an interpretationis echoed by Koechlin, who thought that in all polytonal combinations, one key wasgenerally more strongly projected. This definition of polytonality fits well with PeterKaminsky’s recent identification of primary and secondary tonalities in Ravel.23 It alsohelps to clarify the definition of polytonality cited above, one that is not limited to theprojection of multiple tonalities. Koechlin did admit that in certain polytonal combi-nations, it was quite difficult to determine which key was more strongly projected.24

For example, Igor Stravinsky’s use of the Petrushka chord seems to belong to thiscategory of polytonality.25 Arthur Berger has described tonal deadlock of this sort aspolarity, the necessary conditions of which are “the denial of priority to a single pitch-class precisely for the purpose of not deflecting from the priority of the whole complexesonore.”26 Koechlin’s observation divides polychords into two categories, with amajority that project a single tonality with characteristic dissonances, and a minoritythat project multiple tonalities (Berger’s complexes sonores). While the ratio between

21 There is a similarity between minor-third related triads/keys and Hindemith’s concept of indefinite thirdrelation. The similarities between Hindemith’s theory and polytonality are discussed in greater depth below.22 Specifically, the three notes from G major that are not found in Bb major (F#, B-natural, and E-natural) intro-duce upper chromatic clashes with the dominant, tonic, and subdominant scale degrees, respectively.23 Kaminsky, “Ravel’s Late Music,” 238–248.24 “I readily admit, however, that in many of the harmonically polytonal examples cited above, it is quite difficultto determine which is the primary tonality in them!” Italics in original. Koechlin, “Évolution de l’harmonie,” 723.Koechlin’s belief that one key in a polytonal combination was generally more strongly projected (mentionedabove) is implied in his comment on the exceptional nature of the excerpts he references here.25 Stravinsky’s ballet Petrushka (1911–1912) features a polychord that is associated with the main character. Thepolychord is composed of both C and F# major triads. These six notes belong to a single octatonic (diminished)scale, which serves as the recurrent, associative harmony for Petrushka. For more information on the rich historyof Russian harmonic characterization, see Richard Taruskin, “Chernomor to Kashchei: Harmonic Sorcery orStravinsky’s ‘Angle’,” Journal of the American Musicological Society 38 (1985): 72–142. For more information onStravinsky’s varied use of octatonic harmony throughout his long career, see Pieter van den Toorn, The Music ofIgor Stravinsky (New Haven, CT: Yale University Press, 1983).26 Arthur Berger, “Problems of Pitch Organization in Stravinsky,” Perspectives of New Music 2 (1963): 11–43.

Example 2



Koechlin’s two categories of polytonal combinations may be questioned, his generalobservation provides compelling evidence as to why the identification of polytonalitycannot be linked to the projection of multiple tonalities.

The Dave Brubeck Octet of the 1950s that grew directly from Milhaud’s compositionclass performed a number of compositions that had previously been turned in as thegroup’s homework assignments. One of the earliest of these numbers is Brubeck’sCurtain Music, so named because it was used both to open and close each of the Octet’sshows. Curtain Music is a prime example of the type of melodic polytonality thatBrubeck frequently used in his works for this ensemble. Melodic polytonality inevitablyproduces harmonic complexity, and Curtain Music is no exception. Indeed, this pieceis written in A major, but an unadulterated A major triad appears only once, in thework’s final bar (see Example 3).27

27 This example is a piano reduction made by the present author from the performance found on Dave Brubeck,Dave Brubeck Octet, Fantasy OJCCD – 1012, 1999, compact disc.

Example 3 Dave Brubeck, Curtain Music (1946). Curtain Music by Dave Brubeck, copy-right © 1954 (renewed 1982) Derry Music Company.



In Curtain Music, the tonic function does appear elsewhere beyond the last measure,but only in forms colored by polytonality. For instance, the accented tonic triad thatbegins the work is heard above the subtonic triad as part of a polychord, and the tonictriad on the second beat of the opening measure is embellished with a lowered ninth inthe bass voice. While the tonic/subtonic polychord is used to punctuate downbeats, thetonic chord with a raised ninth serves as the harmonic point of departure and arrivalin the opening two bars, despite the fact that it features a semitonal clash between itsouter voices. Measures 5–11 build on this idea, for when the two hands arrive at theirseparate melodic goals, there is frequently a semitonal clash between them. The reasonfor these clashes is easily discovered: the right hand is written in E major, and only theA# of m. 11 lies outside this scale. The left-hand part contains more chromaticism thanthe right, but its line begins and ends with the top half of a D# scale (though the last twonotes of the final melodic ascent A# – B# – C# – D# in m. 11 are given harmonic supportin a different key). Thus, the two established keys relate to one another by semitone.

If it is remembered that tonic functions in jazz often appear as seventh chords ratherthan simple triads, the semitonal clashes that contribute to the polytonal effect inCurtain Music can be explained in another way. Heard in this light, the melodic goalsof each hand in mm. 5–11 sound on the one hand polytonal, and on the other hand liketonic seventh chords in the local key of E major. In other words, Curtain Music can beheard in a single key, but one sprinkled liberally with “wrong notes.” This description,one that immediately invokes Stravinskian neo-classicism, is chosen quite deliberately,as the first album that Brubeck bought after returning from active duty in World War IIwas notably a recording of Stravinsky’s Pulcinella Suite.28 The music from this balletimmediately inspired several works for the Octet, including Curtain Music as well asPlayland at the Beach and Rondo.29 In these numbers, the entire ensemble projects asingle key, one that is embellished with dissonances formed by individual performers’lines written in a separate, though less strongly projected, tonality.

Brubeck’s compositional style changed between the time of his Octet and the forma-tion of his famous quartet in the 1950s. As opposed to the earlier contrapuntal style ofthe Octet, he began to create polytonality almost exclusively through the superimposi-tion of chords. Due to this change of style, it is necessary first to review a statementthat Milhaud once made concerning harmonic polytonality. Milhaud wrote in hisautobiography that polytonal chords satisfied his ears “more than the normal ones, fora polytonal chord is more subtly sweet and more violently potent.”30 As evidencedhere, he placed polychords on a continuum between consonance and dissonance. Thisperspective is an important point because it allows for the creation of musical motion

28 This was related to the author by Brubeck in a personal conversation from July 2003. In this same conversation,Brubeck referred to his Stravinskian influence as a “bag” from which he could pull things from when composingand performing. In context, the “things” to which Brubeck referred were the Stravinskian techniques he hadlearned over his long years of study of this repertoire. Stravinsky’s influence on Brubeck will be more fully docu-mented in the discussion below relating to the latter composer’s late works.29 A later work to reveal the same influence in both style and name is History of a Boy Scout, modeled on Stravinsky’sL’histoire du soldat.30 Milhaud, My Happy Life, 65.



as dissonant polychords resolve to consonant ones, just as the resolution of seventhchords to triads create musical motion in tonal music. In order to reveal this aspect ofmusical organization, it is necessary to find an analytic tool that allows us to judge therelative dissonance of one polychord when compared with another one.

Example 4 reproduces Milhaud’s exhaustive survey of polyharmonies built frommajor triads.32 Beneath each polychord is its pitch-class set identification and inter-val vector. This information is palindromic, showing that triads juxtaposed byinversionally-related intervals contain the same interval content.33 It may seemstrange to apply pitch-class set theory—an analytic technique developed specificallyfor atonal music—to polychords. In fact, the concepts of inversional equivalence, Zrelations, or any of the other facets of this theory that have been critiqued since itsinception, will not be invoked.34 Instead, only one aspect of this theory, the intervalvector, will be raised since it provides a quick summary of the number of eachinterval contained within a polychord. This highly selective appropriation from set

31 The theoretical terminology here requires a bit of explanation. A pitch-class set is any possible combination ofthe 12 semitones within the octave. When pitch duplication as well as transpositional and inversional equivalencebetween two pitch-class sets are eliminated, there are only 212 distinct sets that contain between 3 and 9 members,and each is named by its cardinality followed by a second numeric label. The Z label for the first and last of thepolychords in Example 4, although meaningful, is not important for the purposes of this study. An interval vectorrepresents a tally of all the interval classes contained within a pitch-class set. There are only six entries in an intervalvector since interval classes, unlike intervals, are inversionally equivalent (ascending and descending semitones—intervals 1 and 11, respectively—both are represented by interval class 1; ascending and descending whole steps—intervals 2 and 10, respectively—both are represented by interval class 2, etc.)32 This example is adapted from Figure 2.1 in Mawer, Darius Milhaud, 20.33 For more information on pitch-class set theory, see: Allen Forte, The Structure of Atonal Music (New Haven:Yale University Press, 1973); John Rahn, Basic Atonal Theory (New York: Schirmer Books, 1980); and Joseph N.Straus, Introduction to Post-Tonal Theory, 2nd ed. (Upper Saddle River, NJ: Prentice Hall, 2000).34 Among the many articles critical of pitch-class set theory, the most important include: William Benjamin,“Ideas of Order in Motivic Music,” Music Theory Spectrum 1 (1979), 23–34; George Perle, “Pitch-Class SetAnalysis: An Evaluation,” The Journal of Musicology 8 (1990), 151–172; Richard Taruskin, “Revising Revision,”dual review of Kevin Korsyn, “Towards a New Poetics of Musical Influence,” and Joseph N. Straus, Remaking thePast: Musical Modernism and the Influence of the Tonal Tradition, in Journal of the American Musicological Society46 (1993), 114–138; and Ethan Haimo “Atonality, Analysis, and the Intentional Fallacy,” Music Theory Spectrum18 (1996), 167–199.

Example 4 Chart reproduced from an example in Darius Milhaud’s “Polytonalité etatonalité” (1923), with additional pitch-class set identification and interval vector31



theory can therefore serve as the basis of a system to identify a polychord’s relativedissonance.

The analytic goal here is similar in intent to Paul Hindemith’s theory of harmonicfluctuation.35 This current study’s methodology, however, relies exclusively on thenumber of dissonant intervals contained within a polychord as revealed by its intervalvector. Further deviation from Hindemith is found in the classification of intervals,where the minor second is labeled here as the most dissonant interval (rather than thetritone). This divergence from Hindemith stems from an observation by Harrison,who, when commenting on the diatonic nature of Milhaud’s melodies and theirsuperimposition, concludes that “T6 can be used to create harmonic effects that havea ‘conservative,’ tonal cast to them, while T1 can create effects of a more ‘radical’kind.”36 I suggest that the superimposition of chords from various diatonic collec-tions follows this same rule; while the Petrushka chord may not have a tonal cast,Milhaud’s chord type I/XI demands resolution to a far greater extent. The remainingdissonant interval (interval class 2/10) is the least salient of all. This hierarchy isreflected in the relative weight assigned to these three dissonant interval classes: thedissonance weight for interval class 2/10 is 0.5; for interval class 6 is 2; and for intervalclass 1/11 is 4.37 The dissonance quotient of any polychord can be easily determined,as it is the sum of the product of multiplying the number of each of the dissonantinterval classes contained within a polychord by the dissonance weight of that intervalclass. Once this quotient has been determined, it can be used to compare the relativedissonance between chords.38

The partitioning of a polychord—as well as the register in which it appears—canaffect its perceived dissonance, although only in rare circumstances do these factorsplay a role in the analytic process. With these caveats in mind, Milhaud’s chord typesVI and I/XI are found to be the most dissonant with quotients of 15 and 14.5, respec-tively. Chord types II/X and IV/VIII both have a quotient of 8, while the least dissonantpolychords are chord types III/IX and V/VII, with quotients of 6.5 and 5, respectively.Although these dissonance weight values were not determined in a completely

35 Paul Hindemith, The Craft of Musical Composition, vol. 1, Theoretical Part, trans. Arthur Mendel (New York:Associated Music, 1942), 115 ff.36 Harrison, “Bitonality,” 401. The theoretical terminology here again requires a bit of explanation. Harrisonis referring to the transposition of pitch-class sets, and in this case, the pitch content of a Milhaud melody. Theletter T refers to the transposition that relates two melodies with the same pitch content together, and the numbers1 and 6 refer to the number of semitones of this transposition (the semitone and tritone, respectively).37 Since tritones are self-inverting, their weighting will appear to be twice of that of the other interval classes.38 A second possible method for calculating relative dissonance involves applying a dissonant weight to each inter-val class, including the consonant interval classes 3–5. Yet difficulties arise with interval class 5 since the perfectfourth could be either a consonance or a dissonance based on context. Because of this (and other difficulties in theassignment of dissonance weights to consonant intervals), only the dissonant intervals are considered in themethod of calculating the dissonant quotient in this study. It is further possible to divide the dissonant quotientof a chord by the number of intervals it contains, thereby using an average of the dissonances rather than theirsum. While this averaging method seems mathematically sound, it produces skewed results. For example,Milhaud’s chord type IV/VIII would have a dissonant quotient average of 4 and chord type VI an average of 2.1.Such results are possible since chord type IV/VIII has no entries for interval classes 2/10 and 6. Thus, the processof averaging the dissonance quotient incorrectly identifies the former chord type as the most dissonant fromExample 4 above.



objective manner, the results seem to make musical sense as chord type VI is seen to bethe most dissonant of Milhaud’s chord types, with chord types I/XI virtually equal insalience. The least dissonant of the polychords is Milhaud’s chord type V/VII, since allthe notes of this polychord belong to the same diatonic collection.

With these various points in mind, a good illustrative example can be found inBrubeck’s 2003 composition, Tonalpoly. While my analysis of Tonalpoly breaks theinitial chronological survey of Brubeck’s works, this recent composition notably limitsitself to the polychords listed in the previous example, aside from Tonalpoly’s finalconsonant sonority. Tonalpoly also provides a prime example of Brubeck’s carefulplacement of polychords to create a sense of musical motion.

As seen in Example 5, there is a consistent use of two-bar phrasing in Tonalpoly, andthe majority of phrases end on polychords with a dissonance weight of between 6.5 and8 (phrases 1–2, 4–7). The first two phrases (mm. 1–2 and 3–4) begin and end withchord type II/X; the motion within these phrases, as well as the others in this work, isshown in the upper level of analytic symbols (an open circle represents relative conso-nance while a closed circle represents relative dissonance; phrases are generally repre-sented by a single motion although exceptions are made when a single motion wouldnot accurately reflect the overall shape of the phrase). Motion created by changes indissonances is therefore felt only within each of these phrases, while the uniformity oftheir cadences sets up larger-scale patterns, to be discussed shortly. The succession ofpolychords in the first phrase present a great diversity of dissonance weights, althoughthis fluctuation is partially offset by its harmonic rhythm, the most rapid of the entirepiece. The second phrase is more uniform in its level of dissonance, although the firstappearance of chord type VI raises the level early in this phrase. Once this polychordappears, it is used as the cadential goal in the third phrase (mm. 5–6). The followingphrase (mm. 7–8) is required to resolve this dissonance; the cadential formula of chordtype VI moving to chord type II/X returns us to the dissonance level of the openingphrases and concludes the opening section of the work. The large-scale motion createdby the cadential sonorities is shown in the second-level analysis, which uses the sameanalytic symbols as described above for the first-level (intraphrase) analysis.39

The dissonance level of the work’s middle section (mm. 9–16) fluctuates greatly asin the first phrase, but not for the same reason: the rhythm of the hands frequentlycreates incomplete polychords which greatly reduces the dissonance level in metricallyweak positions of each beat. In addition to this small-scale fluctuation, there is a generaldecrease in the level of dissonance throughout the first three phrases of this section(mm. 9–10, 11–12, and 13–14). This motion is used to set up the appearance of chordtype VI at the end of the fourth and final phrase of this section (mm. 15–16). After areturn of the work’s opening section, a coda follows that begins with chord type VI anduses it as the penultimate chord as well, making the motion to the final tonic triad allthe more satisfying.

39 The motion from relative dissonance to consonance, or vice versa, is similar to Fred Lerdahl’s relaxing and tens-ing branches. While Lerdahl has explored this analytic territory in tonal and post-tonal music, he has avoided theanalysis of polytonality. See Fred Lerdahl, Tonal Pitch Space (Oxford: Oxford University Press, 2001).



The coda of Tonalpoly seamlessly moves from a polychordal to a tonal vocabularywhen it introduces a major triad as its final sonority. Brubeck employed much moresophisticated techniques for moving between passages of polytonality and a standard

Example 5 Dave Brubeck, Tonalpoly (2003). Tonalpoly by Dave Brubeck, copyright© 2009 Derry Music Company.



jazz vocabulary in the works he wrote for his quartet. A review of three compositions,“Strange Meadowlark,” “The Duke,” and “In Your Own Sweet Way,” reveals Brubeck’smost striking successes in this regard, as well as examples of his handling of dissonancelevels on a larger scale than that found in Tonalpoly.40

Brubeck described “Strange Meadowlark” as “the most conventional song” onhis famous 1959 album, Time Out. Despite this description, “Strange Meadowlark”presents an intricate mixture of polychords and high tertian sonorities that characterizethe standard jazz vocabulary.41 The one polychord that can be heard to project twoseparate tonal centers appears infrequently in this work, and it is consistently of asingle type: a dominant seventh chord with a major triad superimposed above it, theirroots related by major second (see the arpeggiated chord in the first two systems ofExample 6).42 This polychord’s dissonance quotient is equal to Milhaud’s chord typeI/XI and its effect in this work is striking for several reasons. First, this polychord is

40 The Brubeck performances used for the discussion of these works are as follows: “Strange Meadowlark,” fromDave Brubeck Quartet, Time Out, Columbia CK 40585, 1990 (orig. rec. 1959), compact disc; and “The Duke”(orig. rec. 1954), from Dave Brubeck, Dave Brubeck: Greatest Hits, Columbia 32046, 1994, compact disc; and “InYour Own Sweet Way,” from Brubeck Plays Brubeck, Columbia 065772, 1998 (orig. rec. 1956), compact disc. Thisarticle’s transcriptions from these works are by the author, with reference to the published transcriptions byHoward Brubeck that appear in The Dave Brubeck Anthology (Van Nuys, CA: Alfred Publishing, 2005).41 For an attempt to accommodate these high tertian sonorities within a traditional Schenkerian framework, seeSteve Larson, “Schenkerian Analysis of Modern Jazz: Questions about Method,” Music Theory Spectrum 20 (1998):209–41.42 This example has been shortened from the published version for reasons of space. Only minor differencesbetween the two versions have been omitted: the final chord of m. 1 is not arpeggiated when repeated; and the firstchord of m. 8 is an open fifth rather than octave when repeated.

Example 6 Dave Brubeck, “Strange Meadowlark” (1959), mm. 1–20. “Strange Meadow-lark” by Dave Brubeck, copyright © 1960 (renewed 1988) Derry Music Company.



much more dissonant than the high tertian chords that surround it; second, it appearsalmost consistently as a harmonic goal, and forms part of a large-scale reduction indissonance that is felt throughout the first two appearances of the work’s main theme.Each of these factors will be explored below (see Example 6).

Because there is overlap between true polychords and standard voicings of moretraditional jazz chords, virtually every sonority in this excerpt can be spelled as a poly-chord.43 However, most of these harmonies are more readily heard simply as hightertian chords. The opening chord, for example, is partitioned as an Eb major seventhchord in the sustained parts with a Bb major triad arpeggiated above; it is heard,however, as a tonic triad embellished with chordal seventh and ninth in the melody.The dual nature of these high tertian chords allows for the application of dissonance

43 Indeed, jazz theory frequently conceptualizes chords as being composed of superimposed elements. Forinstance, Mark Levine discusses the “sus” chord as a subtonic triad superimposed over the dominant scale degree,and he defines “slash” chords as follows: “the note to the left or above the slash represents a triad and the note tothe right or below the slash represents a bass note, or, as in the next example, another triad. This last exampleshows a B triad over a C triad, usually notated B/C.” Mark Levine, The Jazz Piano Book (Petaluma: Sher Music,1989), 23 and 142, respectively.

Example 6 continued.



weight analysis, which reveals Brubeck’s large-scale organization of dissonance. Thefirst and second points of repose (m. 1, beat 3, and m. 3, beat 3) are each sustained andend on the polychord mentioned above, whose dissonance quotient of 14.5 is far higherthan the preceding chords. The next sustained sonority in m. 5 looks like a polychord,but is actually a dominant ninth chord with lowered fifth. Its quotient of 6 begins areduction of dissonance in the cadential sonorities: the following phrase ends in m. 7on another jazz sonority voiced deceptively as a polychord with a quotient of 3.5. Thefinal phrase of the opening material (mm. 8–10) increases the dissonance level muchin the manner of a half cadence; a continuous reduction in dissonance levels is heardin the repetition of this material, where the final phrase modulates to G and ends on asonority—a major triad with added sixth—that has a quotient of only 0.5.

Like “Strange Meadowlark,” polytonality in “The Duke” (originally recorded in1954) appears within a theme—specifically, the B phrase of the latter work’s AAB songform. Yet while the polytonal opening theme of “Strange Meadowlark” dominates thework, the contrast between the traditional jazz harmony of the repeated A phrase of“The Duke” and the polytonality of its contrasting B phrase is striking (see Example 7).It is this contrast in harmonic language that served as Brubeck’s initial idea, for his orig-inal title was “The Duke Meets Darius Milhaud.” In the opening two two-bar phrasesof the “Milhaud” section of this work (mm. 9–12), the two hands move in contrarymotion and polychords gradually give way to less dissonant cadential sonorities. Theremaining four bars of this theme reverse this process: a sequence based on an embel-lished ii° – V – i progression opens the phrase (in mm. 13–14), followed by contrarymotion between the hands and a gradual increase in dissonance until the finalpolychord is reached. This latter chord (in m. 16) serves as an elegant link back to thetraditional language of the opening “Duke Ellington” section. The root of this chord isDb, and it is possible to hear this chord as a Db13 with raised 11. Because the maintheme that follows this phrase begins with a tonic C chord, the root harmony of this Dbpolychord is also the tritone substitution for the dominant function. Brubeckfrequently uses this type of “pivot” chord—one that can be heard both functionally andas a polychord—in order to transition from a passage of one harmony to the other.44

Another example of this type of pivot polychord is found in “In Your Own SweetWay.” This is one of the earliest works that Brubeck composed for his quartet, and itsoriginal version did not include any polytonal harmony.45 The later solo piano versionof 1956, however, prominently features polytonality, something that the composer/pianist says was not planned for the recording date, but simply “happened.”46 In thisversion, the work’s main theme is initially harmonized in a traditional jazz idiom, aside

44 Before leaving this work, the hidden complexity of its first theme must be mentioned since it reveals the extentto which Brubeck was influenced by the Second Viennese composers. This eight-bar passage begins and ends in Cmajor, but easily moves through a number of keys in between, including D, Eb, Db, Bb, and Ab major. In fact,chords with roots on all twelve chromatic pitches appear in the course of this theme. Brubeck did not discover thisfor himself, but was told by a fan years after he wrote the work. It is for this reason that Brubeck jokingly said thatthe work should be renamed “The Duke Meets Darius Milhaud and Arnold Schoenberg in the Bass Line.”45 See, for example, the performance of this work on Dave Brubeck, Time Signatures: A Career Retrospective(Columbia/Legacy 66047, 2000, compact disc boxed set), that was recorded shortly after its composition.46 This was related by Brubeck to the author in a personal conversation from July 2003.



from the prominent appearance of chord type VI as the second chord (not shown inExample 8 below). This single polychord motivates a polytonal reharmonization of themain theme on its return (shown in Example 8 below). The first two two-bar phrases(mm. 26–29) present an increase in dissonance, while this motion is reversed in the third

Example 7 Dave Brubeck, “The Duke” (1955), mm. 1–16. “The Duke” by Dave Brubeck,copyright © 1955 (renewed 1983) Derry Music Company.



and final four-bar phrase (mm. 30–33). The cadential sonorities of these phrases mirrorthe motion of the final phrase: dissonant chord type VI that ends the first phrase yieldsslightly to the cadential chord of the following phrase and completely to the simple triadthat ends the excerpt. The pivot from polychords to triads takes place in this final phrase.Immediately following two phrases of polychords, the third phrase’s initial sonority—an F dominant seventh chord with lowered fifth in the left hand, with a Db major triadin the right hand—sounds like a polychord, but it can also be heard as an exotic super-tonic, one that leads to an authentic cadence in the tonic key of Eb (see Example 8).

The end of Brubeck’s classic quartet in 1967 gave the composer/pianist more ofan opportunity to write for different musical forces. He did, however, continue tocompose for his own ensembles. Tritonis, from 1978, fits into both of these categories:it was originally commissioned for flute and guitar and was later transcribed for pianoand flute, piano solo, and for jazz quartet. Tritonis is written in 5/4, and each bar isdivided into three beats followed by two. Throughout much of this work, the first threebeats of each measure present the main harmony, while the remaining two beatsconsistently arpeggiate a tritone-related triad. Thus, despite the fact that harmonicmotion in this work moves almost exclusively through the circle of fifths, each barcontains a melodic statement of chord type VI and a harmonic clash on its final twobeats. Tritone substitution could be used to explain this melodic organization;however, because this technique is systematically incorporated into the metric organi-zation of this work, tritone substitution seems an insufficient label here. In fact,because of the melodic, harmonic, and metric formulae, polytonality and functional

Example 8 Dave Brubeck, “In Your Own Sweet Way” (1956), mm. 26–33. “In Your OwnSweet Way” by Dave Brubeck, copyright © 1955 (renewed 1983) Derry Music Company.



tonality in Tritonis are held in equilibrium. Only the cadences—the first two formed bya thinning down to a single instrument (in mm. 10 and 21–22), the third by therepeated statements of chord type VI that resolve to a single triad (in mm. 39–41)—provide relief to this tonal/polytonal stasis (see Example 9).47

According to Brubeck, Tritonis represents “the place I hoped to arrive at whenI started playing. The music is both polytonal and polyrhythmic.”48 The same ideal is

47 The analysis of this work in Example 8 is from the performance on Brubeck’s Time Signatures CD anthology.The transcription of this work is by the author.48 Dave Brubeck, liner notes to Brubeck, Time Signatures, 24.

Example 9 Dave Brubeck, Tritonis (1978), mm. 1–41. Tritonis by Dave Brubeck, copyright© 1979 Derry Music Company.



equally seen in his ballet, Glances (1983), which was composed only a handful of yearslater. The four movements of this ballet seemingly exhibit more diversity in their poly-tonal language than Tritonis, as different key signatures between the two hands are acommon feature of this work. Despite this fact, Brubeck has most recently unequivo-cally stated that Tritonis is the summit of his polytonal explorations.49 An overview ofGlances reveals a possible reason for Brubeck’s assessment: the main themes of both the

49 This was related by Brubeck to the author in a personal conversation from July 2003.

Example 9 continued.



second (“Struttin’”) and fourth (“Doin’ the Charleston”) movements use the superim-position of keys related by minor third (see Example 10). This design is an idea that, asstated above, dates back to the beginning of Brubeck’s career. Furthermore, thisspecific polytonal combination is more susceptible to being heard in a single key, unlikethe polytonal formula heard in Tritonis.50 However, while Glances represents a stepbackwards in terms of its polytonal language when compared to Tritonis, the formerwork does contain more rhythmic complexity.

The most striking aspect of the new rhythmic interplay in Glances is its source, forlike the neo-classical influence on the Octet mentioned above, Brubeck again revealsthe influence of Stravinsky, as I shall demonstrate.51 Changing meter is common inthe work’s overture, yet in certain passages, this changing meter actually hides anunderlying steady meter. Such a passage first appears in m. 26. Here, the time signa-ture of both hands is determined by the changing meter of the right-hand part, whilethe left-hand presents a simple duple-meter pattern that runs counter to the notatedmeter. The changing meter sets key motifs—including the ascending runs of mm. 26and 28, the descending G major arpeggios of mm. 30 and 32, and the descending Gb

50 “Struttin’” was composed earlier than the other movements and was originally entitled “Polly,” which is an allu-sion to the name of a family friend as well as the polytonal language of the work.51 This is not to imply that rhythmic complexity is lacking in any of Brubeck’s jazz influences (including the influ-ences of Cleo Brown, Art Tatum, and Duke Ellington), but only that the rhythmic complexities found in Brubeck’slate works most closely represent those found in Stravinsky’s scores.

Example 10a Dave Brubeck, “Struttin’” from Glances (1983), mm. 59–66. “Struttin”’ fromthe ballet Glances by Dave Brubeck, copyright © 1976 Derry Music Company.



major arpeggiations of mm. 31 and 33—to begin on the downbeat of their respectivemeasures. In relation to the underlying duple meter of the left hand, however, thesesame events switch metric position from on-the-beat to off-the-beat, or vice versa.These measures are shown in Example 11 as notated and also as rebarred according tothe left hand’s regular meter to more clearly reveal the rhythmic displacement that isfelt in the right-hand line.

This type of rhythmic play, which is defined by a reversal of metric placement anddescribed by the theorist and Stravinsky scholar Pieter van den Toorn as a “sadistic

Example 10b Dave Brubeck, “Doin’ the Charleston” from Glances (1983), mm. 1–16.“Doin the Charleston” from the ballet Glances by Dave Brubeck, copyright © 1976 DerryMusic Company.



twitch,” is a hallmark of Stravinsky’s music.52 Indeed, every aspect of the passagedescribed above—the changing meter governed by the melodic line, the underlyingstable meter at odds with the notated meter, the rhythmic immobility of motifs in rela-tion to the notated meter, and the change in rhythmic placement in relation to theunderlying stable meter—can be found (to cite but one example among many) in anexcerpt from Stravinsky’s L’histoire du soldat. As seen in Example 12, this excerpt is

52 Van den Toorn, “Rhythmic (or Metric) Invention,” in The Music of Igor Stravinsky, 204–51.

Example 11 Dave Brubeck, “Overture” from Glances (1983), mm. 26–33. “Overture”from the ballet Glances by Dave Brubeck, copyright © 1976 Derry Music Company.



Example 12 Igor Stravinsky, “Marche du soldat” from L’histoire du soldat (1918),mm. 44–50. Music by Igor Stravinsky. Libretto by Charles Ferdinand Ramuz Music copy-right © 1924, 1987, 1992 Chester Music Ltd., 14–15 Berners Street, London W1T 3LJ, UK,worldwide rights except the United Kingdom, Ireland, Australia, Canada, South Africa,and all so-called reversionary rights territories where the copyright © 1996 is held jointlyby Chester Music Limited and Schott Music GmbH & Co. KG, Mainz, Germany. Librettocopyright © 1924, 1987, 1992 Chester Music Ltd, 14–15 Berners Street, London W1T 3LJ,UK. All rights reserved. International copyright secured. Reprinted by permission.



again presented twice: the first shows this passage in its original notation, while thesecond is rebarred according to the underlying stable meter.53

Brubeck openly acknowledges Milhaud’s influence: he refers to the French composeras his mentor, and he even named one of his sons Darius. More importantly, Brubeck’searly lessons with Milhaud prompted his lifelong exploration of polytonality. Brubeck’sconception of polytonality—arising primarily through the superimposition of triads—has also notably not changed since his early introduction to Milhaud’s article. Thissustained interest in Milhaud’s ideas is a testament to Brubeck’s devotion to his formercomposition teacher; it is also evidence of theoretical convenience, for this approach topolytonality encompasses both polychords and the high tertian chords that characterizethe traditional jazz vocabulary. The analyses in this study show that Brubeck master-fully exploits the overlap between polychords and high tertian chords not only tocontrol the dissonance level within his works, but also to transition from sectionsdevoted to one vocabulary to another. Brubeck’s careful control of the dissonance levelin his works stems not only from the stylistic necessities of the jazz idiom, but also fromthe polytonal works of Milhaud, which reveal this same quality.

While Brubeck has been open about the debt he owes to Milhaud, he has admittedin private to being influenced by Stravinsky.54 The analyses in this study reveal some ofthe ideas that Brubeck has used from his “bag of Stravinsky’s things” that he has accu-mulated. Stravinsky’s early neo-classical works served as audible models for some ofBrubeck’s earliest works. Stravinskian ideas are more fully incorporated into Brubeck’slate style, as in the rhythmic play used in Glances and the conspicuous use of octatonicharmony in Tritonis. Jazz harmony has long recognized the “diminished scale”;Brubeck, however, consciously avoided using this scale until late in his career.55 Theconsistent appearance of the scale in Tritonis56 only years before his adoption of rhyth-mic displacement makes the link with Stravinsky all the more apparent. While each ofthese ideas can be readily found individually in jazz harmony, taken together they pointinstead to a Stravinskian influence. The extent of Brubeck’s debt to both Milhaud andStravinsky has yet to be fully uncovered, yet this study has shown that having fusedclassical and jazz styles, Dave Brubeck owes as much of a debt to both Milhaud andStravinsky than he does to the jazz musicians that preceded him.

53 The first two traits mentioned above are clear in this example, while the second two are more difficult to see(although they are obvious to the ear). The rhythmic immobility is seen in the inner voice motive, in which thenote A3 is consistently notated as an anacrusis. The change is concerned with rhythmic placement and is heard inthe two appearances of this same note, which, when renotated, appears first as an anacrusis and then as adownbeat.54 This was related by Brubeck to the author in a personal conversation from July 2003. In fact, Brubeck remem-bers stopping dead in his tracks as he was walking across campus as an undergraduate and heard the universityorchestra rehearsing Stravinsky’s Symphony of Psalms (this was this first time he had heard the work). This eventhappened at approximately the same time as Brubeck first read Milhaud’s article on polytonality, so he was intro-duced to both these composers at roughly the same time in his musical development.55 The standard use of this scale in jazz—as a descending scalar run over a dominant seventh chord—appearsprominently in “Eleven Four” from 1962, although this tune was written by Paul Desmond.56 See, for instance, the passages (m. 27 and mm. 39–41) labeled as octatonic in Example 9.



Acknowledgement

Preliminary research for this project was made possible through a grant from South-eastern Louisiana University.

Abstract

Dave Brubeck has incorporated polytonality into his jazz compositions throughouthis long career. Like his composition teacher Darius Milhaud, Brubeck defines polyto-nality as the combination of distinct triads, and this technique forms the definition ofthe term as used in this article. This approach avoids the insoluble problems of chordspelling and perception inherent in polytonality; it also allows for a grey area betweensimple polychords and the projection of multiple tonal centers (and Brubeck exploitsboth procedures in his compositions).

This article introduces a method to compare the relative dissonance between poly-chords in order to reveal the logic behind Brubeck’s incorporation of polytonality intothe standard jazz vocabulary. Brubeck’s use of polytonality helps to project a generaldecrease or increase in relative dissonance, thereby clarifying the formal structure onboth the small- and large-scale. The comparison with tonal theory extends to includepivot chords; with Brubeck, such chords simultaneously serve as the final chord in apolychordal passage and as the first and most exotic chord in a tonal passage.

The final goal of this article is to trace Brubeck’s influences. Milhaud is the mostobvious of these, but certain Stravinskian features are also found in Brubeck’s music,including rhythmic practices first identified by the theorist Pieter van den Toorn.


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